Class 11 Mathematics Trigonometric Equations MCQs Set 03

Question. The number of points of intersection of the two curves \( y = 2\sin x \) and \( y = 5x^2 + 2x + 3 \) is
(a) 0
(b) 1
(c) 2
(d) \( \infty \)
Answer: (a) 0

Question. A solution (x,y) of \( x^2 + 2x \sin(xy) + 1 = 0 \) is
(a) \( (1,0) \)
(b) \( (1,7\pi / 2) \)
(c) \( (-2,7\pi / 2) \)
(d) \( (-1,0) \)
Answer: (b) \( (1,7\pi / 2) \)

Question. The solution set of equation \( \cos^5 x = 1 + \sin^4 x \) is
(a) \( \{n\pi, n \in I\} \)
(b) \( \{2n\pi, n \in I\} \)
(c) \( \{4n\pi, n \in I\} \)
(d) \( \{n\pi / 2, n \in I\} \)
Answer: (b) \( \{2n\pi, n \in I\} \)

Question. Number of solution (s) of the equation \( \sin x = [x] \) [where \( [\bullet] \) denotes greatest integer function is
(a) 1
(b) 2
(c) 0
(d) infinite
Answer: (b) 2

Question. The roots of the equation \( \cos^7 x + \sin^4 x = 1 \) in the interval \( (-\pi, \pi) \) are
(a) \( \left\{ -\frac{\pi}{2}, 0, \frac{\pi}{2} \right\} \)
(b) \( \left\{ -\frac{\pi}{2}, \frac{\pi}{2} \right\} \)
(c) \( \left\{ \frac{\pi}{2} \right\} \)
(d) \( \{\pi\} \)
Answer: (a) \( \left\{ -\frac{\pi}{2}, 0, \frac{\pi}{2} \right\} \)

Question. Let n be a positive integer such that \( \sin \frac{\pi}{2n} + \cos \frac{\pi}{2n} = \frac{\sqrt{n}}{2} \) Then
(a) n = 6
(b) n = 1,2,3,....8
(c) n = 5
(d) n = 4
Answer: (a) n = 6

Question. If \( x = X \cos \theta - Y \sin \theta, y = X \sin \theta + Y \cos \theta \) and \( x^2 + 4xy + y^2 = AX^2 + BY^2, 0 \leq \theta \leq \frac{\pi}{2}, n \in Z \), then .
(a) \( \theta = \frac{\pi}{6} \)
(b) \( \theta = \frac{\pi}{4} \)
(c) A = 3
(d) Both 2 and 3
Answer: (d) Both 2 and 3

Question. \( \sin \theta = \frac{1}{2} \left( \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} \right) \) necessarily implies
(a) x > y
(b) x < y
(c) x = y
(d) both x and y are purely imaginary
Answer: (c) x = y

Question. The equation \( 2 \cos^2 \left( \frac{x}{2} \right) \sin^2 x = x^2 + \frac{1}{x^2}, 0 \leq x \leq \frac{\pi}{2} \) has
(a) one real solution
(b) no solution
(c) more than one real solution
(d) Two solutions
Answer: (b) no solution

Question. Let \( 2\sin^2 x + 3 \sin x - 2 > 0 \) and \( x^2 - x - 2 < 0 \) (x is measured in radians ). Then x lies in the interval
(a) \( \left( \frac{\pi}{6}, \frac{5\pi}{6} \right) \)
(b) \( \left( -1, \frac{5\pi}{6} \right) \)
(c) \( (-1, 2) \)
(d) \( \left( \frac{\pi}{6}, 2 \right) \)
Answer: (d) \( \left( \frac{\pi}{6}, 2 \right) \)

Question. If \( \sin^4 x + \cos^4 y + 2 = 4 \sin x \cos y \) and \( 0 \leq x, y \leq \frac{\pi}{2} \) then \( \sin x + \cos y \) is equal to
(a) - 2
(b) 0
(c) 2
(d) 5
Answer: (c) 2

Question. If \( 0 \leq x \leq 2\pi \) and \( |\cos x| \leq \sin x \), then
(a) \( x \in \left[ 0, \frac{\pi}{4} \right] \)
(b) \( x \in \left[ \frac{\pi}{4}, 2\pi \right] \)
(c) \( x \in \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right] \)
(d) \( [0, \pi] \)
Answer: (c) \( x \in \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right] \)

Question. Number of ordered pairs (a,x) satisfying the equation \( \sec^2(a+2)x + a^2 - 1 = 0; -\pi < x < \pi \) is
(a) 2
(b) 1
(c) 3
(d) infinite
Answer: (c) 3

Question. The number of the solutions of the equation \( \cos(\pi \sqrt{x-4}) \cos(\pi \sqrt{x}) = 1 \) is
(a) > 2
(b) 2
(c) 1
(d) 0
Answer: (c) 1

Question. The most general values of \( \theta \) for which \( \sin \theta - \cos \theta = \min_{a \in R}(1, a^2 - 6a + 11) \) are given by
(a) \( n\pi + (-1)^n \frac{\pi}{4} - \frac{\pi}{4}, n \in I \)
(b) \( n\pi + (-1)^n \frac{\pi}{4} + \frac{\pi}{4}, n \in I \)
(c) \( 2n\pi + \frac{\pi}{4}, n \in I \)
(d) \( n\pi + \frac{\pi}{2}, n \in I \)
Answer: (b) \( n\pi + (-1)^n \frac{\pi}{4} + \frac{\pi}{4}, n \in I \)

Question. The number of values of x in \( [0, 2\pi] \) satisfying the equation \( |\cos x - \sin x| \geq \sqrt{2} \), is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. In \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), \( \log_{\sin \theta}(\cos 2\theta) = 2 \) has
(a) no solution
(b) a unique solution
(c) two solutions
(d) infinetly many solutions
Answer: (b) a unique solution

Question. Let [x] = the greatest integer less than or equal to x and let \( f(x) = \sin x + \cos x \). Then the most general solution of \( f(x) = \left[ f \left( \frac{\pi}{10} \right) \right] \) are
(a) \( 2n\pi \pm \frac{\pi}{2}, n \in Z \)
(b) \( n\pi, n \in Z \)
(c) \( 2n\pi - \frac{\pi}{2}, n \in Z \)
(d) \( 2n\pi \text{ or } 2n\pi + \frac{\pi}{2} \)
Answer: (d) \( 2n\pi \text{ or } 2n\pi + \frac{\pi}{2} \)

Question. Consider the solutions of the equation \( \sqrt{2} \tan^2 x - \sqrt{10} \tan x + \sqrt{2} = 0 \) in the range \( 0 \leq x \leq \pi/2 \). Then only one of the following is true
(a) no solutions exist for x in the given range
(b) two solutions \( x_1 \) and \( x_2 \) exist with \( x_1 + x_2 = \pi/4 \)
(c) two solutions \( x_1 \) and \( x_2 \) exist with \( x_1 + x_2 = \pi/6 \)
(d) two solutions \( x_1 \) and \( x_2 \) exist with \( x_1 + x_2 = \pi/2 \)
Answer: (d) two solutions \( x_1 \) and \( x_2 \) exist with \( x_1 + x_2 = \pi/2 \)

Question. The number of solutions of the equation \( |\cos x| = 2[x] \) is (where \( [\bullet] \) denotes the reatest integer function )
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0

Question. If \( \alpha, \beta \) are solutions of \( \sin^2 x + a\sin x + b = 0 \) and \( \cos^2 x + c\cos x + d = 0 \) then \( \sin(\alpha + \beta) \) equals
(a) \( \frac{2ac}{a^2 + c^2} \)
(b) \( \frac{a^2 + c^2}{2ac} \)
(c) \( \frac{2bd}{b^2 + d^2} \)
(d) \( \frac{b^2 + d^2}{2bd} \)
Answer: (a) \( \frac{2ac}{a^2 + c^2} \)

Question. If \( \sqrt{3} \sin \pi x + \cos \pi x = x^2 - \frac{2}{3}x + \frac{19}{9} \), then x is equal to
(a) \( -\frac{1}{3} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{4}{3} \)
Answer: (b) \( \frac{1}{3} \)

Question. The equation \( 1 + \sin^2 ax = \cos x \) has a unique solution then a is
(a) rational
(b) irrational
(c) integer
(d) whole number
Answer: (b) irrational

Question. If n be the number of solutions of the equation \( |\cot x| = \cot x + \frac{1}{\sin x} (0 < x < 2\pi) \), then n =
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. If \( \frac{1 - \tan x}{1 + \tan x} = \tan y \) and \( x - y = \frac{\pi}{6} \), then x,y are respectively
(a) \( \frac{5\pi}{24}, \frac{\pi}{24} \)
(b) \( -\frac{7\pi}{24}, -\frac{11\pi}{24} \)
(c) \( -\frac{115\pi}{24}, -\frac{119\pi}{24} \)
(d) All of the options
Answer: (d) All of the options

Question. If \( \theta \in [0, 5\pi] \) and \( r \in R \) such that \( 2 \sin \theta = r^4 - 2r^2 + 3 \) then the maximum number of values of the pair \( (r, \theta) \) is
(a) 8
(b) 10
(c) 6
(d) 4
Answer: (c) 6

Question. If \( \frac{\sin^3 \theta - \cos^3 \theta}{\sin \theta - \cos \theta} - \frac{\cos \theta}{\sqrt{1 + \cot^2 \theta}} - 2 \tan \theta \cot \theta = -1, \theta \in [0, 2\pi] \) then
(a) \( \theta \in \left( 0, \frac{\pi}{2} \right) - \left\{ \frac{\pi}{4} \right\} \)
(b) \( \theta \in \left( \frac{\pi}{2}, \pi \right) - \left\{ \frac{3\pi}{4} \right\} \)
(c) \( \theta \in \left( \pi, \frac{3\pi}{2} \right) - \left\{ \frac{5\pi}{4} \right\} \)
(d) \( \theta \in (0, \pi) - \left\{ \frac{\pi}{4}, \frac{\pi}{2} \right\} \)
Answer: (d) \( \theta \in (0, \pi) - \left\{ \frac{\pi}{4}, \frac{\pi}{2} \right\} \)

Question. The least difference between the roots, in the first quadrant \( \left( 0 \leq x \leq \frac{\pi}{2} \right) \), of the equation \( 4 \cos x (2 - 3 \sin^2 x) + (\cos 2x + 1) = 0 \), is
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{2} \)
Answer: (a) \( \frac{\pi}{6} \)

Question. The set of all x in \( (-\pi, \pi) \) satisfying \( |4\sin x - 1| < \sqrt{5} \) is given by
(a) \( \left( -\frac{3\pi}{10}, \frac{3\pi}{10} \right) \)
(b) \( \left( -\frac{\pi}{10}, \pi \right) \)
(c) \( (-\pi, \pi) \)
(d) \( \left( -\pi, \frac{3\pi}{10} \right) \)
Answer: (a) \( \left( -\frac{3\pi}{10}, \frac{3\pi}{10} \right) \)

Question. The general solution of the equation \( \frac{1 - \sin x + .... + (-1)^n \sin^n x + .....}{1 + \sin x + .... + \sin^n x + .......} = \frac{1 - \cos 2x}{1 + \cos 2x} \) is
(a) \( (-1)^n \left(\frac{\pi}{3}\right) + n\pi, \forall n \in I \)
(b) \( (-1)^n \left(\frac{\pi}{6}\right) + n\pi, \forall n \in I \)
(c) \( (-1)^{n+1} \left(\frac{\pi}{6}\right) + n\pi, \forall n \in I \)
(d) \( (-1)^{n-1} \left(\frac{\pi}{3}\right) + n\pi, \forall n \in I \)
Answer: (b) \( (-1)^n \left(\frac{\pi}{6}\right) + n\pi, \forall n \in I \)

Question. The value of x between 0 and \( 2\pi \) which satisfy the equation \( \sin x \sqrt{8\cos^2 x} = 1 \) are in AP with common difference
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{8} \)
(c) \( \frac{3\pi}{8} \)
(d) \( \frac{5\pi}{8} \)
Answer: (a) \( \frac{\pi}{4} \)

Question. The no. of values of x in the interval \( [0, 3\pi] \) satisfying the equation \( 2\sin^2 x + 5\sin x - 3 = 0 \) is
(a) 1
(b) 4
(c) 6
(d) 2
Answer: (b) 4

Question. The solution set satisfying \( \tan x > 1 \) is
(a) \( \left( n\pi + \frac{\pi}{4}, n\pi + \frac{\pi}{2} \right) \)
(b) \( \left( n\pi + \frac{\pi}{4}, n\pi \right) \)
(c) \( \left( n\pi + \frac{\pi}{4}, \infty \right) \)
(d) \( \phi \)
Answer: (a) \( \left( n\pi + \frac{\pi}{4}, n\pi + \frac{\pi}{2} \right) \)

Question. \( 0 \leq a \leq 3, 0 \leq b \leq 3 \) and the equation, \( x^2 + 4 + 3\cos(ax + b) = 2x \) has atleast one solution then the value of a + b
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( \pi \)
Answer: (d) \( \pi \)

Question. If \( 2\tan^2 x - 5\sec x = 1 \) for exactly 7 distinct values of \( x \in \left[ 0, \frac{n\pi}{2} \right], n \in N \) then the greatest value of n is
(a) 13
(b) 17
(c) 19
(d) 15
Answer: (d) 15

Question. If the equation \( \cot^4 x - 2\csc^2 x + a^2 = 0 \) has at least one solution, then the sum of all possible integral values of a is equal to
(a) 4
(b) 3
(c) 2
(d) 0
Answer: (d) 0

Question. The equation \( \sin^4 x + \cos^4 x + \sin 2x + \alpha = 0 \) is solvable for
(a) \( -\frac{5}{2} \leq \alpha \leq \frac{1}{2} \)
(b) \( -3 \leq \alpha \leq 1 \)
(c) \( -\frac{3}{2} \leq \alpha \leq \frac{1}{2} \)
(d) \( -1 \leq \alpha \leq 1 \)
Answer: (c) \( -\frac{3}{2} \leq \alpha \leq \frac{1}{2} \)

Question. The number of solution of \( \sum_{r=1}^5 \cos rx = 5 \) in the interval \( [0, 2\pi] \) is
(a) 0
(b) 1
(c) 5
(d) 2
Answer: (d) 2

Question. The number of solution of \( x \in [0, 2\pi] \) for which \( [\sin x + \cos x] = 3 + [-\sin x] + [-\cos x] \) (where [.] denotes the greatest integer function) is
(a) 0
(b) 4
(c) infinite
(d) 1
Answer: (c) infinite

Question. The equation \( \cos^8 x + b\cos^4 x + 1 = 0 \) will have a solution if b belongs to
(a) \( (-\infty, 2] \)
(b) \( [2, \infty) \)
(c) \( (-\infty, -2] \)
(d) \( [1, \infty) \)
Answer: (c) \( (-\infty, -2] \)

Question. Assertion (A): \( 3\sin x + 4\cos x = 7 \) has no solution
Reason (R): \( a\cos x + b\sin x = c \) has no solution if \( |c| > \sqrt{a^2 + b^2} \)
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true and R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is correct explanation of A

Question. Statement I: Principal value of \( \cos\theta = -1 \) is \( \pi \)
Statement II: Principal value of \( \sin\theta = 0 \) is \( \pi \)
Which of the above statement is correct?
(a) Only I
(b) Only II
(c) Both I and II
(d) Neither I or II
Answer: (a) Only I

Question. Statement I: The set of values of x for which \( \frac{\tan 3x - \tan 2x}{1 + \tan 3x \tan 2x} = 1 \) is \( \left\{n\pi + \frac{\pi}{4}, n \in I\right\} \)
Statement II: The expression \( (1 + \tan x + \tan^2 x)(1 - \cot x + \cot^2 x) \) is positive for all defined values of x
III) \( e^{\sin x} - e^{-\sin x} \neq 4 \) for any real values of x.
Which of the following is correct?
(a) only I, II
(b) only II, III
(c) only I, III
(d) I, II, III
Answer: (b) only II, III

Question. Statement I: If \( \cot\left(\frac{\pi}{3}\cos 2\pi x\right) = \sqrt{3} \), then the general solution of the equation is \( x = n \pm \frac{1}{6}, n \in I \).
Statement II: If \( \tan P\theta = \tan Q\theta \), then the values of \( \theta \) from an A.P with common difference \( \frac{\pi}{P - Q} \).
Which of the above statements are correct?
(a) only I
(b) only II
(c) Both I and II
(d) neither I or II
Answer: (c) Both I and II

Question. Statement I: If x lies in the 1st Quadrant and \( \cos x + \cos 3x = \cos 2x \) then \( x = 30^\circ \) or \( 45^\circ \)
Statement II: \( x \in (0, 2\pi) \) and \( \text{cosec } x + 2 = 0 \) then \( x = \frac{7\pi}{6}, \frac{11\pi}{6} \)
Statement III: \( x \in [0, 2\pi] \) and \( (2\cos x - 1)(3 + 2\cos x) = 0 \) then \( x = \frac{\pi}{3}, \frac{5\pi}{3} \)
Which of the above statements are correct?
(a) only I, II
(b) only II, III
(c) only I, III
(d) I, II, III
Answer: (b) only II, III